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Maximum distance profile codes over finite non-binary fields have been introduced and thoroughly studied in the last decade. These codes have the property that their column distances are maximal among all codes of the same rate and degree. In this paper, we aim at studying this fundamental concept in the context of convolutional codes over a finite ring. We extensively use the concept of $p$ -encoder to establish the theoretical framework and derive several bounds on the column distances. In particular, a method for constructing (not necessarily free) maximum distance profile convolutional codes over ${\mathbb Z}_{p^{r}}$ is presented. [ABSTRACT FROM AUTHOR]

Maximum distance separable (MDS) convolutional codes are characterized through the property that the free distance meets the generalized Singleton bound. The existence of free MDS convolutional codes over $${\mathbb {Z}}_{p^{r}}$$ was recently discovered in Oued and Sole (IEEE Trans Inf Theory 59(11):7305-7313, 2013) via the Hensel lift of a cyclic code. In this paper we further investigate this important class of convolutional codes over $${\mathbb {Z}}_{p^{r}}$$ from a new perspective. We introduce the notions of p-standard form and r-optimal parameters to derive a novel upper bound of Singleton type on the free distance. Moreover, we present a constructive method for building general (non necessarily free) MDS convolutional codes over $${\mathbb {Z}}_{p^{r}}$$ for any given set of parameters. [ABSTRACT FROM AUTHOR]